Signed Euler characteristic conjecture for Kähler manifolds with nonpositive bisectional curvature

Let $M$ be an $n$-dimensional compact Kähler manifold with nonpositive bisectional curvature, and suppose that its Ricci curvature is quasi-negative. Its signed Euler characteristic is the number $(-1)^n c_n[M]$, where $c_n[M]$ denotes the top Chern number of $M$.

Signed Euler characteristic conjecture. Under these hypotheses,

$$(-1)^n c_n[M]>0.$$

The source describes this as a complex analogue of the Hopf conjecture. It records that the claim is known when $n=2$, and when $n\leq 4$ under the stronger assumption that the Kähler metric has nonpositive Riemannian sectional curvature; the full statement remains unresolved in the source.